Nowhere dense in a metric space

Show that

A subset A of a metric space X is nowhere dense in X if and only if each non-empty open set in X contains an open ball whose closure is disjoint from A.

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Some definitions
1. A subset A of a metric space X is nowhere dense in X if has empty interior.
2. Let A be a subset of a metric space X. A point x in A is an interior point of A provided that there is an open set O which contains x and is contained in A.

A subset A of a metric space X is nowhere dense in X if and only if each non-empty open set in X contains an open ball whose closure is disjoint from A.

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Some definitions
1. A subset A of a metric space X is nowhere dense in X if has empty interior.
2. Let A be a subset of a metric space X. A point x in A is an interior point of A provided that there is an open set O which contains x and is contained in A.

NOTE: You may want to wait for a more senior knowledgable member to either validate or correct my suggestion.

This is almost by defintion. Supose the opposite that is dense in and that the hypothesis holds...use this to show that not every point of is a limit point of

A subset A of a metric space X is nowhere dense in X if and only if each non-empty open set in X contains an open ball whose closure is disjoint from A.

Is it still true if the word "closure" is taken out such that
"A subset A of a metric space X is nowhere dense in X if and only if each non-empty open set in X contains an open ball which is disjoint from A"
If it is wrong, any counterexample?